Calibration method, device and computer program

ABSTRACT

A method for calibrating parameters of sensor elements in a sensor array. The method comprises receiving an output signal of at least two sensor elements signal in reaction to an input signal from a signal source; estimating a cross-correlation between the output signals of at least two of said sensor elements; and optimising a difference between the estimated cross-correlation and a cross-correlation model; and thereby estimating said parameters from the optimised difference. A cross-corelation model is used as represented by the mathematical equation: R=G B G H +D

TECHNICAL FIELD AND BACKGROUND OF THE INVENTON

The invention relates to a method and a device for calibratingparameters of sensor elements in a sensor array. The invention alsorelates to a computer program product for calibrating parameters ofsensor elements of a sensor array when run on a programmable device andto a sensor array calibrated with a calibration method.

In sensor array systems, the, complex, receiver gains and sensor noisepowers of the sensor elements in the sensor array are initially unknownand have to be calibrated. (Gain) calibration enhances the quality,specifically the sensitivity, of the sensor system and, moreover,improves the effectiveness of array signal processing techniques forinterference mitigation.

Non-polarized and single polarization calibration techniques for sensorarray systems are generally known in the art [1,2,3,5,13], and thestatistical performance also is a well studied topic [1,2,3,5,8,13,14].Recently, Hamaker, Bregman and Sault [10,11,12] developed, for radioastronomical purposes, a polarization formalism in which thepolarization state of the received signal, and the propagation of thesesignals through the atmosphere and through the sensor array, werethoroughly and elegantly incorporated. This formalism is based on optics[15,16] and on extensions of the (approximate) solutions in radiopolarimetry [17,18].

In this formalism, the polarized signal is described by a 2×2 sizeStokes matrix [17,18] (a Stokes matrix describes the polarization stateof the signal: intensity, linearity, ellipticity, polarization angle,total polarization), and the distorting and propagation effects by a 2×2size Jones matrix [10,11,12] which in general is different for each ofthe dual polarization array sensors. The output of a dual polarizationchannel is described by a multiplication of Jones and Stokes matrices.The polarized array formalism is further focussed on pair-wisecorrelation products involving 2×2 size Jones and Stokes matrices.However, solving systems based on this formalism require an iterativeapproach and convergence is not always guaranteed. Hence, in suchsystems stability is a problem as well.

The single polarization and non-polarized sensor array parameterestimation problem is well known from literature [2,13]. However, thesecalibration methods are disadvantageous because they require a largeamount of processing. They also require a good initial point (gain andnoise values), which is not always available. Typically, the number ofprocessing steps involved scales with the third power of the number ofsensor elements.

Recently, fast and closed form single polarization calibrationtechniques were described in [1]. In this publication, the calibrationtechniques involves the comparison of an estimated signal with a signaloutputted by telescopes in a telescope array. By optimisation of theestimated signal with a least square error minimalisation of thedifference between the estimated and outputted signal, the gains of thetelescopes can be derived. In the publication [1] several variants ofthe least square error minimalisation are described. One of theminimalisations is a logarithmic minimalisation, in which difference ofthe logarithms of the covariance of the estimated signals and thecovariance of the outputted signals are compared.

The number of processing steps for the logarithmic minimalisationdescribed in [1] scales with the square of the number of elements and isthus much much faster than conventional methods. However thislogarithmic minimalisation has the disadvantage that, for unequal gains,the method is not efficient, which means that the estimation accuracy islower than the theoretical bound.

SUMMARY OF THE INVENTION

It is a goal of the invention to provide a better calibration method,more specific it is a goal of the invention to provide a numericallystable calibration method by providing a method for which closed formsolutions exist for non-polarized, single polarization, and dualpolarization sensors.

Therefore, the invention provides a calibration method according toclaim 1.

For a method according to the invention closed form estimation solutionsexist. Thereby, the parameters can be estimated in a single and stableprocess.

In one embodiment, a method according to the invention is applied to thecalibration of a dual polarised sensor array, that is an array for thereception of dual polarised signals. For polarised signals inparticular, a method according to the invention gives new insights inthe gain estimation issue, specifically, new (closed form) solutions tothe estimation problem now become available. Furthermore, a dualpolarization calibration method according to the invention has theadvantage that closed form solutions exist. Such a system is numericallymore stable than the known calibration systems

In another embodiment, a method according to the invention is applied toa single polarization or a non-polarized sensor array and a weightedlogarithmic minimalisation is used to estimate the parameters. It can beshown, that this embodiment, for low SNR, is asymptotically efficient.Thus, for a large number of data samples, the estimator is efficient forlow SNR. The improvement is larger for larger gain differences, thus theembodiment is especially suited for sensor arrays with large unequalgain magnitude differences.

Furthermore, the invention provides a calibration system according toclaim 17. Also, the invention provides a calibrated sensor arrayaccording to claim 20.

The invention also provides a computer program product according toclaim 21. Such a computer program enables a programmable device tocalibrate a sensor array in a stable manner when the program is run onthe programmable device.

Specific embodiments of the invention are set forth in the dependentclaims. Further details, aspects and embodiments of the invention willbe described with reference to the attached drawing.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically shows an example of a first embodiment of a dualpolarization calibration system according to the invention.

FIG. 2 schematically shows an example of second embodiment of a singlepolarization or non-polarized calibration system according to theinvention.

FIG. 3 schematically shows a flowchart of an example of a calibrationmethod according to the invention.

FIG. 4 schematically shows a flowchart of an example of a calibrationmethod according to the invention.

DETAILED DESCRIPTION

In the following description of examples of embodiments of theinvention, the following notation is used. Bold capital letters indicatematrices, bold lowercase letters represent vectors, and non-bold letters(either lowercase or uppercase) indicate scalars. Matrix and vectorelements are denoted by subscripts. The subscript n denotes the n-thobserved time sample. Superscript * denotes the complex conjugate.Superscript t denotes the vector or matrix transpose, which is anoperation which switches the columns and rows of the matrix or vector,i.e. ((R^(t))_(ij)=R_(ji)). Superscript H denotes the Hermitian(conjugate) transpose of a matrix ((R^(H))_(ij)=((R*^(t))_(ij)=R*_(ji)).Re{ } represents the real part, Im{ } the imaginary part and i is thesquare root of −1. E{ } denotes the expected value of the covariancematrix. x_(n) denotes the output vector at time n. Dual polarizationvectors and matrices are represented by slanted letters; for the singlepolarization case non-slanted letters are used. The number of single ordual polarization array sensors is denoted by p.

EXAMPLE A

Polarization Calibration

In the following, by way of example, a gain calibration system and amethod according to the invention are described in a radio-astronomyapplication. However, the invention is by no means limited toapplications in radio-astronomy and may likewise be applied in otherfields, such as for example array signal processing systems in which a,dominant, point like emitting source with an arbitrary polarizationstate is present, such as for example satellite tracking phased arraysor otherwise. The method is based on observations of three separate orconsecutive (polarized or non-polarized) point-like signal sources.

For one signal source, R represents the ‘true’ or expected 2p×2pcovariance matrix (of the 2p×1 dual polarization array output vectorx(t)=(x₁ ^(x)(t), x₁ ^(y)(t), . . . , x_(p) ^(x)(t), x_(p) ^(y)(t)),thus:R=E{x(t)x(t)^(H)}  (A1)

The polarization properties of the source [15,16,17,18] are representedby a 2×2 (Stokes) matrix B, the 2p×2p noise matrix D is defined byD=E{d(t)d(t)^(H)}, and d(t)=(d₁ ^(x)(t), d₁ ^(y)(t), . . . , d_(p)^(x)(t), d_(p) ^(y)(t)), where d_(i) ^(x)(t) and d_(i) ^(y)(t) indicatethe sensor noise. D may be either diagonal (in case of a perfectinsulation between the two polarizations of a dual sensor) or blockdiagonal (in case of a leakage between the two polarizations of a dualsensor). G represents the polarization gain matrix of dimensions 2p×2.

FIG. 1 shows a dual polarization sensor array with 2 p sensors elementsby way of example of a system 1 according to the invention. The system 1is able to perform the example of a method according inventionillustrated in FIG. 2. In FIG. 1, three point-like sources 2 ₁, 2 ₂, 2₃, emit consecutively each a signal 3 ₁, 3 ₂, 3 ₃ to the system 1. Instep 101 in FIG. 3, each of the three consecutive signals 3 ₁, 3 ₂, 3 ₃is observed separately by a number of dual polarization sensor elements4 ₁,4 ₂, . . . ,4 _(p) (p representing the total number of dualpolarization sensor elements) of a sensor array 4, which may for example(but not necessarily) be an uniform linear sensor array. The outputs ofthe sensor elements 4 ₁,4 ₂, . . . ,4 _(p) are connected to inputs 51 ofa gain calibration device 5. The outputs of the sensor array arefurther, as indicated with the striped lines to a, not shown, beamformerdevice as is for example known in the arts of radars, acoustic arrays,and radio-astronomy.

The output signals x(t) contain the input signal s(t) of a single sourcemultiplied with a (complex) number α_(τ) which represents the path timedelay for the respective sensor elements 4 _(i). The signal s(t) alsomultiplied with a gain factor G_(i) of the respective sensor element 4_(i). The output signal x_(i)(t) also comprises system noise d_(i)(t)added in each sensor element 4 _(i) to the signal s(t).

Each dual polarization sensor 4 ₁,4 ₂, . . . ,4 _(p) comprises twosensor elements, one for each polarization component in the signals 3₁-3 ₃. Thus, the total number of sensors elements in the system 1 is 2p.In steps 102 and 103, the output signals x_(i) ^(x)(t) resp. x_(i)^(y)(t) are presented at the outputs of the dual polarization arraysensor elements 4 ₁,4 ₂, . . . , 4 _(p), where x^(x) is the outputsignal for one polarization component of the input signal, and x^(y) isthe output signal for the other polarization component of the inputsignal.

In the described examples, it is assumed that the look direction, i.e.the direction of the source 2 with respect to the sensor array 4, isknown. Thus, without loss of generality, the path time delay a_(τ) canbe set to 1, e.g. the delay is taken to be the same for all sensorelements. However, the invention may likewise be applied if the pathtime delay differs for some or all sensor elements. The inputs 51 arecommunicatively connected to a vectoriser device 52, In step 104 theoutput signals x_(i)(t) are stacked in the output vector x(t)=(x₁^(x)(t), x₁ ^(y)(t), . . . , x_(p) ^(x)(t), x_(p) ^(y)(t)) by thevectoriser device 52.

Connected to the vectoriser device 52 is a correlator device 53 which isable to cross-correlate, in step 105 in FIG. 3, elements in the outputvector x(t). In the shown correlator device 53, a covariance matrix isformed by determining the covariance of the element outputs x_(i)(t).However other correlation methods may be used as well. After steps101-105, the gain parameters, as represented by a matrix G of size 2p×2(except for an arbitrary phase offset term), are estimated in steps 106and further, by solving cost functions, each cost function correspondingto a different one of the input signals 3 ₁, 3 ₂, 3 ₃.

The calibration device 5 further comprises a estimator device 54connected to the cross-correlator device 53, which is able to estimatean estimated cross-correlation In step 106, the estimatedcross-correlation matrix R_(est) is estimated by the estimator 54 fromthe covariance matrix established by correlator device 53. However, theestimator device 54 may likewise estimate the cross-correlation in adifferent manner

In step 107, an optimisation device 55 communicatively connected to theestimator device 54, compares the estimated covariance matrix R_(est)with a model of the covariance matrix R stored in a memory 56.

In the example, in step 107 a difference between the cross-correlationmodel and the estimated cross-correlation matrix R_(est) is optimised,by minimising the difference, which difference is also referred to asthe estimation error. In step 108 the gain parameters are estimated orderived from the result of step 107. In the example of FIG. 3, in step106 the gain parameters are derived from the optimised differencebetween the estimated matrix and the model by a gain estimator device57. The estimator device 57 is connected to the optimisation device 55.The estimator device 57 is also connected to an calibrator output 58,via which the estimated parameters can be transmitted further to be usedin further processing such as the processing of signals received at thesensor elements after the calibration.

In the example of a device of FIG. 1 and the example of a method of FIG.3, as a cross-correlation model, a model of the covariance matrix R isused. The covariance matrix R₁ for one source can be expressed as:R ₁ =G B ₁ G ^(H) +D  (A2)and for Q sources the covariance matrix R can be expressed as:R={σ _(m=1 . . . Q) G B _(m) G ^(H) }+D.  (A3)

Using the model of equations A2 and/or A3, given an estimate covariancematrix R_(est), for one or more consecutive sources, the gain matrix Gand noise matrix D can be estimated. In this example, in step 106 anestimate of the covariance matrix R_(est) is obtained by the estimatordevice 54 by forming the product:R _(est)=(1/N)σ_(n) x(t _(n))x(t _(n))^(H)  (A4)

Thus, in this example A, the estimated covariance matrix R_(est) is atime averaged covariance matrix of the output signals. However, it islikewise possible to estimate the covariance matrix in a differentmanner.

It can be shown [12] that for a system of three differently polarizedsources an unambiguous or unique gain solution can be obtained (apartfrom an arbitrary phase offset valid for the entire system). Thepolarized gain can be found by solving in steps 107 and 108 theequation: $\begin{matrix}{\left\{ {G,D} \right\} = {\underset{G,D}{\arg\quad\min}\left\{ {\left( {{R_{{est},1} - \left\{ {{{GB}_{1}G^{H}} + D} \right\}}}_{F} \right)^{2} + \left( {{R_{{est},2} - \left\{ {{{GB}_{2}G^{H}} + D} \right\}}}_{F} \right)^{2} + \left( {{R_{{est},3} - \left\{ {{{GB}_{3}G^{H}} + D} \right\}}}_{F} \right)^{2}} \right\}}} & ({A5})\end{matrix}$

Here the subscript F means the Frobenius norm, which is the square rootof the sum of the squares of all matrix elements. Thus, the added leastsquare differences between the estimated covariance matrices for outputsignals resulting from input signals from three sources and thecorresponding covariance matrix models are optimised. However, it islikewise possible to optimise the difference in a different manner. Theestimated gain and noise parameters of the sensor elements are thenestimated to be the gain and noise parameters for which equation (A5) issolved.

As an example of the solution of eq. (A5), a rank two factor analysisapproach may be used, as is briefly described below. However, other moreor less standard solutions for the equation (A5) exist and the inventionis not limited to the example described below. Solutions for equationssimilar to eq. (A5) are generally known in the art. The example methodstarts with solving:R ₁ =G B ₁ G ^(H) +D  (A6)for the estimated covariance matrix R_(est,1) of the first signal.

This can be carried out by applying a rank two factor analysis [6] tosubmatrices M of R_(est,1) (both M and R_(est,1) are rank two), in whichthe main (block) diagonal of R_(est,1) is not included as it containsthe system noise as well. A resulting (rank 2) matrix A-can be formed,based on the obtained M's, with the property that, for the off (block)diagonal elements, AA^(H) is equal to R. D can be obtained bycalculating R_(est,1)−AA^(H).

The next step is to calculate the matrix root of B so that equation [A6]can be expressed as:R=(GB ^(1/2))(GB ^(1/2))^(H) +D  (A7)

With the factor analysis approach, described above, a GB^(1/2) factorcan be found. The problem here is that the solution is not unique as anyunitary matrix can be inserted in equation A7 (between the two GB^(1/2)factors) without affecting R. It can be shown that equations likeequations A6 and A7 can be fully solved (up to a single arbitrary phaseoffset) by using three subsequent, independent, observations (R_(est,1),R_(est,2), and R_(est,3)) of emitting point sources with differentpolarization states using a generalized eigenvalue analysis and thepseudo inverse.

EXAMPLE B

Gain Calibration by Weighted Logarithmic Estimation

In FIG. 2, an example of a system 1 according to the invention is shown.In FIG. 2, a point-like source 2 emits a signal 3 to the system 1. Thesystem 1 comprises sensor elements 4 ₁,4 ₂, . . . ,4 _(p) (prepresenting the total number of sensor elements) of a sensor array 4,for example (but not necessarily) a uniform linear array. The sensorelements receive an input signal s(t) which stems from the signal 3. Atthe outputs of each of the array sensor elements 4 ₁,4 ₂, . . . ,4 _(p)an output signal x_(i)(t) is presented. The outputs of the sensorelements 4 ₁,4 ₂, . . . , 4 _(p) are connected to a gain calibrationdevice 5.

The output signals x_(i)(t) contain the input signal s_(i)(t) multipliedwith a (complex) number a, which represents the path time delay for therespective sensor element 4 _(i). The signal s_(i)(t) is also multipliedwith a gain factor g_(i) of the respective sensor element 4 _(i). Theoutput signal x_(i)(t) also comprises system noise d_(i)(t) added ineach sensor element 4 _(i) to the signal s_(i)(t).

The gain calibration device 5 is able to perform the example of a methodaccording to the invention of FIG. 4. The calibration device 5 has anumber of inputs 51 connected to the outputs of the sensor elements 4₁,4 ₂, . . . ,4 _(p). In step 201, the output signals x_(i)(t) arereceived from the sensor elements. The inputs are communicatively to avectoriser device 52 in which in step 202 the output signals x_(i)(t)are stacked in a vector, denoted by bold lowercase letters: x(t)=[x₁(t),. . . , x_(p)(t)]^(t).

In the following, it is assumed that the look direction, i.e. thedirection of the source 2 with respect to the sensor array 4, is known.Thus, without loss of generality the path time delay a_(τ) can be set to1, e.g. the delay is taken to be the same for all sensor elements.However, the invention may likewise be applied if the path time delaydiffers for some or all sensor elements. The array output vector x(t),under the assumption mentioned above, can be described as x(t)=gs(t)+d(t), where d(t) is a vector containing the array noise signalsd₁,d₂, . . . ,d_(p) and where g denotes the (complex) array gain factorsg₁,g₂, . . . , g_(p) of the sensor elements 4 ₁,4 ₂, . . . ,4 _(p).

Connected to the vectoriser device 52 is a correlator device 53 which isable to cross-correlate, as in step 203 in FIG. 4, the elements in theoutput vector x(t). In the shown correlator device 53 a covariancematrix is formed by determining the covariance of the element outputsx_(i)(t). However other cross-correlation methods may be used as well.The true covariance matrix R is the expected value of the covariancematrix, thus:R=E{x(t)x(t)^(H)}  (B1)

The calibration device 5 further comprises a estimator device 54, whichis able to estimate an estimated cross-correlation, as in step 204 inFIG. 4. In the shown example, the estimated covariance matrix R_(est) isestimated by taking the, weighted, time-average of the covariance matrixof the output signals x_(i)(t) from the correlator device 53. However,the covariance matrix can likewise be estimated in a different manner.In the shown example, the estimated covariance matrix R_(est) canmathematically be described as:R _(est)=(1/N)Σ_(n) x(t _(n))x(t _(n))^(H)  (B2)

In which equation (B2), N represents the total number of the outputvector x(t). Since the noise is uncorrelated to the source signal, thecovariance matrix R can be modelled, see reference [1], as:R=g g ^(H) +D  (B3)

Here, the noise contribution is represented by a diagonal matrix D withthe noise variance on the main diagonal (D=E{d d^(H)}). Thus, from theestimated covariance matrix R_(est), an estimated gain g_(est) and noiseD_(est) can be deduced, because R_(est)=g_(est) g_(est) ^(H)+D_(est). Inthis model [1] it is assumed that the source power is unity.

The estimated gain and estimated noise are then derived by minimising instep 205 and 206 by an optimisation device 55 communicatively connectedto the estimator device 54 and a memory device 56 in which the model isstored, a difference between the estimated covariance matrix R_(est) andthe model of the covariance matrix R, which difference is also referredto as the estimation error. A generally used approach thereto is tominimize the least squares (LS) cost function:{g _(est) ,D _(est) }=argmin _(g,D)(∥R _(est)−(g g ^(H) +D)∥_(F))²  (B4)where subscript F denotes the Frobenius matrix norm. In [1] theminimization is done by solving{g _(est) }=argmin _(g,k)(∥J vec(ln(R _(est))−ln(g g ^(H))+2πki)∥_(F))²where J is a selection matrix which puts zeros on the main diagonalelements, thereby removing the system noise contributions, where k is aphase unwrapping vector containing integer values, and where vec( ) is amatrix operator stacking the matrix elements in a vector. Once the gainsgest are found, the noise matrix is found byD _(est) =R _(est) −g _(est) g _(est) ^(H)  (B5)

In the example of FIGS. 2 and 4, a weighted logarithmic least squarecost function (WLOGLS) is used. The WLOGLS is equal to:{g _(est) }=argmin _(g,k)(∥W J vec(ln(R _(est))−ln(g g^(H))+2πki)∥_(F))²  (B6)

W represents a weighting matrix equal to (D^(−1/2) Γ)⊕(D^(−1/2) Γ), inwhich Γ is defined as diag(|g₁|, . . . |g_(p)|), i.e. Γ is a diagonalmatrix with values of the diagonal elements set to the absolute value ofthe, estimated, gain factors g₁, g₂, . . . , g_(p). Here ⊕ denotes theKronecker matrix product. For a low Signal to Noise Ratio (SNR), the SNRis substantially equal to g g^(H)/trace(D), it can be proved (not shownhere) that the gain estimation is asymptotically efficient, which meansthat the estimation accuracy, for a large number of observed samples,meets the theoretical bound (i.e. it means that it theoretically can notbe estimated more accurately)

To obtain a fast and reliable estimate, the least square optimisationcan be performed in a number of times. In the shown example, theoptimisation is performed two times in steps 205 and 206. In step 205,an estimate is obtained of the gain g and noise D by observing the pointsource and using the identity matrix I as a weight matrix. In step 206,the gains are estimated using the weight matrix constructed as describedabove using the estimated gain g_(est) and noise D_(est) estimated instep 205.

The calibration method of this example B works especially well for lowSNR and also for arrays in which the antenna gains have a wide gainmagnitude variation. The latter is also useful for gain estimation inthe case part of the antennas (due to for example malfunctioning) havevery low SNR.

A method or device according to the invention may be applied in anyarray signal processing system, such as systems in which a dominantpoint-like emitting source is present. For example, a method or a deviceaccording to the invention can be used in cellular telephone basestations, phased array antennas or otherwise. Also, the invention may beapplied for calibration of a directional hearing device, comprising anarray of microphones which can for example be phase-tuned to receiveacoustical signals from a certain direction and reject or filter outsignals from other directions. Furthermore, the signals to be sensed bythe sensor array may be of any suitable type, such as for example radiosignals, acoustical signals, optical signals or otherwise. Likewise, thesignal source may be of any suitable type, such as a satellite in orbitaround a celestial body or a pulsar. In general, pulsars are extremelyprecise pulsating celestial bodies and hence with a sensor arraydirected to a pulsar a precise time-measuring device can be obtained.

Furthermore, the invention can likewise be applied as a data carriercomprising data representing a computer program product, comprisingprogram code for performing steps of a method according to the inventionwhen run on a programmable device. Such a data carrier can for examplebe a read only memory compact disk or a signal transfer medium, such asa telephone cable or a wireless connection. The programmable device maybe of any suitable type. For example, it may be a computercommunicatively connected to a senor array. However, the computer maylikewise be not connected to a sensor array, but receive datarepresenting signals from the array, e.g. via a floppy disk or a compactdisk.

It should be noted that the above-mentioned embodiments illustraterather than limit the invention, and that those skilled in the art willbe able to design may alternatives without departing from the scope ofthe appended claims. In the claims, any reference signs placed betweenparentheses shall not be construed as limiting the claim. The word‘comprising’ does not exclude the presence of other elements or stepsthan those listed in a claim. The mere fact that certain measures arerecited in mutually different claims does not indicate that acombination of these measures cannot be used to advantage. Furthermore,if a document is referenced to in this application, this does notindicate that the document relates to the same field of technology asthe present invention.

REFERENCES

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1. A method for calibrating parameters of sensor elements in a sensorarray, comprising: receiving an output signal of at least two sensorelements signal in reaction to an input signal from a signal source;estimating a cross-correlation between the output signals of at leasttwo of said sensor elements; optimising a difference between theestimated cross-correlation and a cross-correlation model; and therebyestimating said parameters from the optimised difference; wherein across-correlation model is used as represented by the mathematicalequation:R=G B G ^(H) +D in which equation: R represents a cross-correlationmatrix, G represent a gain matrix comprising gain parameters, G^(H)represents the Hermitian conjugate of the gain matrix, D represents a((block) diagonal) noise matrix comprising noise parameters and Brepresents a matrix comprising information about the signal source.
 2. Amethod as claimed in claim 1, wherein said difference is a least squaredifference.
 3. A method as claimed in claim 1, wherein thecross-correlation is obtained by determining a time-averaged covariancematrix from the output signals.
 4. A method as claimed in claim 1,wherein the sensor array is a single polarization or non-polarizedsensor array.
 5. A method as claimed in claim 1, wherein the sensorelements are dual polarization sensor elements for receiving a dualpolarised signal.
 6. A method as claimed in claim 1, wherein said methodis performed for output signals of the sensor elements generated inreaction to input signals from at least three signal sources withdifferent polarizations.
 7. A method as claimed in claim 4, wherein saidoptimising comprises: minimising a difference between a weightedlogarithm of the estimated cross-correlation and a weighted logarithm ofthe cross-correlation and estimating the gain of at least one of thesensor elements from said difference.
 8. A method as claimed in claim 7,wherein the logarithm is weighted by a weighting matrix with matrixvalues relating to said gain parameters.
 9. A method as claimed in claim7, wherein said optimising and said estimating gain parameters areperformed at least a first time and a second time, wherein in the firsttime an uniform weight is used for all output signals and in the secondtime the weight is used in dependence on the gain estimated in the firsttime for the respective output signals.
 10. A method as claimed in claim7, wherein said optimising comprises an operation as represented by themathematical equation:{g _(est) }=argmin _(g,k)(∥W J vec(ln(R _(est))−ln(g g^(H))+2πki)∥_(F))² , in which equation: g_(est) represents the parameterto be estimated; g represents a variable; g^(H) represents the Hermitianconjugate of the variable; J represent a selection matrix which putszeros on the main diagonal; k represents a phase unwrapping vectorcontaining integer values; W represents a weighting matrix; and R_(est)represents the estimated cross-correlation.
 11. A method as claimed inclaim 1, wherein the signal source is a satellite in orbit around acelestial body.
 12. A method as claimed in claim 1, wherein the signalsource is a pulsar.
 13. A method as claimed in claim 1, wherein theoutput signals have a low signal to noise ratio.
 14. A method as claimedin claim 1, wherein the sensor elements are antennas in a phased arrayantenna.
 15. A method as claimed in claim 1, wherein the sensor elementsare electro-magnetic sensors elements.
 16. A method as claimed in claim1, wherein the sensor elements are acoustical sensor elements.
 17. Acalibration system for calibrating parameters of sensor elements in asensor array, comprising at least two inputs, each connectable to anoutput of an sensor element in a sensor array; a correlation estimatordevice for estimating a correlation between the output signals of atleast two of said sensor elements an optimiser device for optimising adifference between the estimated cross-correlation and across-correlation model and thereby estimating said parameters from theoptimised difference; a memory device containing the cross-correlationmodel, which model is represented by the mathematical equation:R=G B G ^(H) +D in which equation: R represents a cross-correlationmatrix, G represent a gain matrix comprising gain parameters, G^(H)represents the Hermitian conjugate of the gain matrix, D represents anoise matrix comprising noise parameters and B represents a matrixcomprising information about the signal source and.
 18. A calibrationsystem as claimed in claim 17, wherein the sensor array is a dualpolarised sensor array.
 19. A calibration system as claimed in claim 17,wherein the sensor array is a single polarization or non-polarizedsensor array.
 20. An array signal processing system calibrated with amethod as claimed in claim
 1. 21. A computer program product, comprisingprogram code for performing steps of a method as claimed in claim 1 whenrun on a programmable device.
 22. A data carrier comprising datarepresenting a computer program product as claimed in claim 21.